Repetition Avoidance in Circular Factors

Abstract

Abstract. We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.

Repetition avoidance in products of factors

“…Proof of RT i (3) 3i 2 + 1 4 for every even i. Mousavi and Shallit [5] have proved that RT 2 (3) = 13 4 , which settles the case i = 2. We have double checked their computation of the lower bound RT 2 (3) 13 4 .…”

Theoretical and Practical Aspects Related to the Avoidability of Patterns in Words

“…Dejean [Dej72] proved RT(3) = 7 4 and conjectured RT(4) = 7 5 and RT(k) = k k´1 for k ą 4. Mousavi and Shallit [MS13] extended this notion to RTC -the circular repetition threshold -and they proved RTC(2) = 4 and RTC(3) = 13 4 . Moreover they established in [MS13] the notion RT i (k) describing the repetition threshold over a k-letter alphabet and concatenating i P N factors.…”

“…Mousavi and Shallit [MS13] extended this notion to RTC -the circular repetition threshold -and they proved RTC(2) = 4 and RTC(3) = 13 4 . Moreover they established in [MS13] the notion RT i (k) describing the repetition threshold over a k-letter alphabet and concatenating i P N factors. They proved RT 2 (k) = RTC(k) for k = 2, 3 and conjectured RT 2 (k) = RTC(k) for k ą 3.…”

“…An extension to this generalisation is to allow the concatenation of the remaining factors in an arbitrary order: if we take mi and imi of minimisation and concatenate them to imimi we get a 5 2 -power of im. Mousavi and Shallit [MS13] investigated the adjusted definition of the critical exponent, i.e. the supremum of exp(u) over all factors u of conjugates of factors of the word, where only two factors are allowed to be concatenated.…”

“…Mousavi and Shallit [MS13] have considered the binary alphabet and obtained that RT i (2) = 2i for every i ě 1. Our second result considers the ternary alphabet and gives the value of RT i (3) for every i ě 1.…”